System and methods of regularized optimization for matrix factorization and image and video reconstruction

ABSTRACT

An image and video processing system and methods based on amplitude-modulation frequency-modulation (“AM-FM”) demodulation to provide high quality reconstructions, both visually and quantitatively. The system and methods reconstructs an image based on a Regularized Optimization (“RO”) of estimates to attain a small number of locally coherent components and simultaneously enforce a piecewise smooth constrain for one or more amplitude functions.

FIELD OF THE INVENTION

The present invention relates generally to image and video processing.More particularly, the present invention is directed to a system andmethods of optimizing Amplitude-Modulation Frequency-Modulation(“AM-FM”) demodulation for processing stationary and non-stationaryimage and video content.

AM-FM demodulation is useful in a variety of contexts and applicationsincluding, for example, characterization and classification of image andvideo from imaging modalities such as electron microscopy, spectral andhyperspectral devices, ultrasound, magnetic resonance imaging (“MRI”),positron emission tomography (“PET”), histology, color and monochromeimages, molecular imaging, radiographs (“X-rays”), computer tomography(“CT”), and others. The specific applications are in fingerprintidentification, detection and diagnosis of retinal disease, malignantcancer tumors, cardiac image segmentation, atherosclerosischaracterization, brain function, histopathology specimenclassification, characterization of anatomical structure tracking suchas carotid artery walls and plaques or cardiac motion and as the basisfor computer-aided diagnosis to name a few.

BACKGROUND OF THE INVENTION

Image and video processing are forms of signal processing. Signalprocessing allows a set of characteristics or parameters related to theimage or video to be obtained. Signal processing including analog signalprocessing, discrete time signal processing, and digital signalprocessing, which may involve a one-dimensional (“1D”), two-dimensional(“2D”) or three-dimensional (“3D”) input signal to which signalprocessing techniques are applied.

Signal processing techniques include transform-based processing such asdiscrete or integral transforms which were implemented prior to AM-FMprocessing. As an example, a 1D analysis of transform-based processingincludes the use of short-time Fourier Transform (“STFT”) fornon-stationary signals. When using STFT, the fast Fourier Transform(“FFT”) of different time intervals of the signals is used to determinethe frequency and phase content. Thus, the STFT is a convenient 2Drepresentation that provides frequency content information at differenttime intervals. A disadvantage is that the STFT cannot be effectivelygeneralized to images and videos. For example, using STFT for imageswould produce a four-dimensional (“4D”) representation and using STFTfor video would produce a six-dimensional (“6D”) representation.

The discrete Wavelet Transform (“DWT”) has also been used fortransform-based image processing. Unlike Fourier Transforms, WaveletTransforms are based on specific functions defined at different scalesand durations. Thus, the DWT is a space-frequency representation of theinput signal and it is related to harmonic analysis is as in FourierTransform. While FFT uses equally spaced frequency division, DWT useslogarithmic divisions of the frequency. A disadvantage is that DWT doesnot measure frequency content directly.

The development of accurate methods for estimating amplitude-modulationfrequency-modulation image decompositions is of great interest due to ispotentially significant impact on image analysis applications includingin the areas of signal, image and video processing. Applications insignal processing include speech signal analysis. Image processingapplications include shape from shading, image pattern analysis, imageinterpolation, fingerprint classification, image retrieval in digitallibraries, image segmentation, and damaged image texture repairs.Applications in video processing include cardiac image segmentation,motion estimation, and motion reconstruction, to name a few.

A number of techniques exist to reconstruct an image from its AM-FMrepresentation in terms of amplitude, phase and frequency functions.Reconstruction of an image involves estimating or computing theamplitude, phase and frequency components of the signals emerging fromeach filter channel and using these components to create an AM-FMrepresentation that best approximate the original image signal.Generally, the more components and channels used, the more informationis recovered, the better the image signal is restored and the better theimage is regenerated. If every component of every channel is used, thiswill yield to the best reconstruction of the original image. However,such approaches lead to very redundant representations that can lead tovery inefficient applications in image analysis. Thus, the goal of anefficient reconstruction process is select few channels and componentsthat best approximates the image signal.

The AM-FM Dominant Component Analysis (“DCA”) and Channelized ComponentAnalysis (“CCA”) are methods used that consist of applying a filterbankto the Hilbert-transformed image, and then applying AM-FM demodulationof each bandpass filtered image. Using DCA, every pixel deliversestimated modulating functions corresponding to the AM-FM component thatis locally dominant at that pixel. Using CCA, a filterbank partitionsthe image into components on a spatially global basis. Each resultingAM-FM component is restricted to lie in a single channel over the entireimage domain. With CCA, the number of components in the computed imagemodel is necessarily equal to the number of channels in the filterbank.AM-FM reconstructions based on the CCA use a reasonably small number oflocally coherent components. In contrast, those based on the DCA onlyuse one component—the estimates from the channel with the maximumamplitude estimate. A disadvantage of DCA and CCA are that they areknown to produce noticeable visual artifacts.

Optimizing the quality of an AM-FM reconstruction image is important dueto the potentially significant impact on various applications. Thus,there is demand for high quality reconstructions in both stationary andnon-stationary processing for use in a variety of contexts andapplications. The present invention satisfies this demand.

SUMMARY OF THE INVENTION

The present invention provides a system and methods of high qualityreconstructions, both visually and quantitatively, when compared tostandard reconstructions using various prior art techniques such asDominant Component Analysis (“DCA”) and Channelized Component Analysis(“CCA”). The present invention is based on a Regularized Optimization(“RO”) to attain a small number of locally coherent components andsimultaneously enforce a piecewise smooth constrain for one or moreamplitude functions. In one embodiment, the small number of locallycoherent components and piecewise smooth constrain for one or moreamplitude functions is based on the estimates from the CCA. Imagecontent from image signal components is obtained from processing aHilbert-transformed image through a filter bank. More particularly, thepresent invention provides a Regularized Optimization (“RO”) method ofreconstructing an image by applying an amplitude-modulationfrequency-modulation (“AM-FM”) demodulation process to an image.

Although the present invention is discussed herein with respect totwo-dimensional (“2D”) images, signals, and digital videos, it iscontemplated the present invention can be extended to any dimensional(“ND”) images, signals and digital videos including three-dimensional(“3D”).

The AM-FM representation of images permits non-stationary image contentto be modeled in terms of amplitude and phase functions using thefollowing equation:

$\begin{matrix}{{b(\xi)} = {\sum\limits_{n = 1}^{L}{{\alpha_{n}(\xi)}{\cos( {\varphi_{n}(\xi)} )}}}} & (1)\end{matrix}$Where b(ξ):

→

is the input image, ξ=(ξ₁,ξ₂)ε

, Lε

, α_(n):

→[0,∞)⁻, and φ_(n):

→

. The interpretation of Equation (1) suggest that the L AM-FM componentimages α_(n)(ξ)·cos(φ_(n)(ξ)), model the essential image modulationstructure, the amplitude functions α_(n)(ξ) model stow-changing imageintensity variations, and the FM components cos(φ_(n)(ξ)) capturecast-changing image intensity variations.

It is contemplated that Equation (1) can also be interpreted as aseparation of texture—FM components cos(φ_(n)(ξ))—from piecewise smoothcontent—amplitude functions α_(n)(ξ)—in an image.

The AM-FM Dominant Component Analysis (“DCA”) and Channelized ComponentAnalysis (“CCA”) consist of applying a collection offilterbanks—bandpass filters—to the original input image. The AM-FMdemodulation of each bandpass filtered image provides estimations ofinstantaneous amplitude (“IA”) functions α_(n)(ξ), instantaneous phase(“IP”) functions φ_(n)(ξ), and instantaneous frequency (“IF”) functionsω_(n)(ξ)=∇φ_(n)(ξ).

The goal using CCA is to obtain a reasonably small number of locallycoherent components such as modeling the input image as in Equation (1).The goal using DCA is to select the estimates from the channel with themaximum amplitude estimate using one component—the dominant component—tomodel the input image.

According to the present invention, the minimum of the following:

$\begin{matrix}{{{J( {a,\zeta} )} = {{\frac{1}{p}{{{f( {a,\zeta} )} - b}}_{p}^{p}} + {\lambda_{a}{T(a)}} + {\lambda_{\zeta}{\zeta }_{1}}}},{{s.t.a} \geq 0},{{\zeta } \leq 1}} & (2)\end{matrix}$attains a small number of locally coherent components and simultaneouslyenforces a piecewise smooth constrain for α_(n)(ξ).

As used herein a_(n), ζ_(n) and b are one-dimensional (“1D”) vectorsrepresenting the two-dimensional (“2D) instantaneous amplitude functionα_(n)(ξ), the two-dimensional function cos(φ_(n)(ξ)) and the image b(ξ).In Equation (2), a equals

$\lbrack \begin{matrix}{a_{1}^{T},} & {a_{2}^{T},} & \ldots & { a_{L}^{T} \rbrack^{T},\zeta}\end{matrix} $equals

$\lbrack \begin{matrix}{\zeta_{1}^{T},} & {\zeta_{2}^{T},} & { {\ldots\mspace{11mu},\mspace{11mu}\zeta_{L}^{T}} \rbrack^{T},}\end{matrix} $and f(a,ζ) equals

$\sum\limits_{n = 1}^{L}{{{diag}( a_{n} )}*{\zeta_{n}.}}$The TV regularization generalization to vector-valued images withcoupled channels is

${T(a)} = {\frac{1}{q}{{\sqrt{{\sum\limits_{n}( {D_{x}a_{n}} )^{2}} + ( {D_{y}a_{n}} )^{2}}}_{q}^{q}.}}$Horizontal discrete derivative operators are represented by D_(x) andvertical discrete derivative operators are represented by D_(y).

The present invention is useful in a variety of contexts andapplications, and allows the identification of disease at differentstages, such as retinal disease (diabetic retinopathy, age-relatedmacular degeneration, glaucoma, etc.), pulmonary diseases(pneumoconiosis, lung nodules tumors, etc.), breast cancer, cellularabnormalities, or any pathological structure in a medical or biomedicalimage or video.

The present invention and its attributes and advantages will be furtherunderstood and appreciated with reference to the detailed descriptionbelow of presently contemplated embodiments, taken in conjunction withthe accompanying Figures.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates a block diagram of one embodiment of AM-FMreconstruction according to the present invention;

FIG. 2 illustrates a block diagram of an exemplary computer system forimplementing the methods according to the present invention;

FIG. 3 illustrates an input image and reconstructions using the CCAmethod, the DCA method, and the method according to the presentinvention;

FIG. 4 illustrates an input image and reconstructions using the CCAmethod, the DCA method, and the method according to the presentinvention; and

FIG. 5 illustrates an input image and reconstructions using the CCAmethod, the DCA method, and the method according to the presentinvention.

DETAILED DESCRIPTION OF EMBODIMENTS OF THE INVENTION

FIG. 1 illustrates a block diagram of AM-FM reconstruction 100 accordingto the present invention. Specifically, FIG. 1 illustrates the solutionto the minimization problem:

$\begin{matrix}{{{J( {a,\zeta} )} = {{\frac{1}{p}{{{f( {a,\zeta} )} - b}}_{p}^{p}} + {\lambda_{a}{T(a)}} + {\lambda_{\zeta}{\zeta }_{1}}}},{{s.t.\; a} \geq 0},{{\zeta } \leq 1}} & (2)\end{matrix}$

An input image signal is provided at step 110, although it iscontemplated that an input video may also be provided. Enforcing twoconstrains in the AM-FM reconstruction, a small number of locallycoherent components and a piecewise smooth constrain for the amplitudefunctions b(ξ) in the following equation:

$\begin{matrix}{{b(\xi)} = {\sum\limits_{n = 1}^{L}{{a_{n}(\xi)}{\cos( {\varphi_{n}(\xi)} )}}}} & (1)\end{matrix}$is desired.

An extended analytic signal of the input image is computed at step 120by applying a Hilbert transform to form a 2D extension analytic signalb(ξ) of the 1D analytic signal. The extended analytic signal isprocessed through the filterbank and data is interpreted at step 130.

The functional J(a,ζ) is convex in a or ζ; but is not necessarily convexin both variables together. In one embodiment of the invention, thenumerical value of the local minimum may be solved by an optimizationprocedure shown in step 140 for which updates are alternated for eachindependent variable.

In one embodiment, the optimization procedure shown in step 140 issummarized by the following Equation (3) and Equation (4), for k=1, 2, .. . :

$\begin{matrix}{\zeta^{(k)} = {{\min\limits_{{\zeta } \leq 1}{\frac{1}{p\;\zeta}{{{f( {a^{({k - 1})},\zeta} )} - b}}_{P\;\zeta}^{P\;\zeta}}} + {\lambda_{\zeta}{\zeta }_{1}}}} & (3) \\{a^{(k)} = {{\min\limits_{a \geq 0}{\frac{1}{p_{\alpha}}{{{f( {a,\zeta^{(k)}} )} - b}}_{p\;\alpha}^{p\;\alpha}}} + {\lambda_{a}{T(a)}}}} & (4)\end{matrix}$to solve Equation (2). Specifically, ζ^((k)) of Equation (3) can besolved based on the non-negative quadratic programming optimizationalgorithm and the FOCUSS algorithm. a^((k)) of Equation (4) can besolved using the vector valued IRN-NQP (vv-IRN-NQP) algorithm which isbased on the non-negative quadratic programming optimization algorithmand on the iteratively reweighted norm (“IRN”) algorithm forvector-valued images.

Solving for ζ^((k)) of Equation (3) and a^((k)) of Equation (4) toultimately solve for Equation (2) is strongly dependent on the accuracyof the initial instantaneous amplitude estimates, for example, a⁽⁰⁾ isthe instantaneous amplitude estimate from CCA. In one embodiment of theinvention, the instantaneous amplitude estimate is obtained by theQuasi-Local Method (“QLM”) and is preferred over the Quasi-EigenApproximately (“QEA”) method since the instantaneous estimates using QLMare less sensitive to perturbations—i.e., noise—than the instantaneousamplitude estimates of QEA. After optimization at step 140, the image isreconstructed at step 150.

FIG. 2 illustrates an exemplary computer system 200, or networkarchitecture, that may be used to implement the methods according to thepresent invention. One or more computer systems 200 may carry out themethods presented herein as computer code. One or more processors, suchas processor 204, which may be a special purpose or a general-purposedigital signal processor, is connected to a communicationsinfrastructure 206 such as a bus or network. Computer system 200 mayfurther include a display interface 202, also connected tocommunications infrastructure 206, which forwards information such asgraphics, text, and data, from the communication infrastructure 206 orfrom a frame buffer (not shown) to display unit 230. Computer system 200also includes a main memory 205, for example random access memory(“RAM”), read-only memory (“ROM”), mass storage device, or anycombination thereof. Computer system 200 may also include a secondarymemory 210 such as a hard disk drive 212, a removable storage drive 214,an interface 220, or any combination thereof. Computer system 200 mayalso include a communications interface 224, for example, a modem, anetwork interface (such as an Ethernet card), a communications port, aPCMCIA slot and card, wired or wireless systems, etc.

It is contemplated that the main memory 205, secondary memory 210,communications interface 224, or a combination thereof function as acomputer usable storage medium, otherwise referred to as a computerreadable storage medium, to store and/or access computer software and/orinstructions.

Removable storage drive 214 reads from and/or writes to a removablestorage unit 215. Removable storage drive 214 and removable storage unit215 may indicate, respectively, a floppy disk drive, magnetic tapedrive, optical disk drive, and a floppy disk, magnetic tape, opticaldisk, to name a few.

In alternative embodiments, secondary memory 210 may include othersimilar means for allowing computer programs or other instructions to beloaded into the computer system 200, for example, an interface 220 and aremovable storage unit 222. Removable storage units 222 and interfaces220 allow software and instructions to be transferred from the removablestorage unit 222 to the computer system 200 such as a program cartridgeand cartridge interface (such as that found in video game devices), aremovable memory chip (such as an EPROM, or PROM) and associated socket,etc.

Communications interface 224 allows software and instructions to betransferred between the computer system 200 and external devices.Software and instructions transferred by the communications interface224 are typically in the form of signals 225 which may be electronic,electromagnetic, optical or other signals capable of being received bythe communications interface 224. Signals 225 are provided tocommunications interface 224 via a communications path 226.Communications path 226 carries signals 225 and may be implemented usingwire or cable, fiber optics, a phone line, a cellular phone link, aRadio Frequency (“RF”) link or other communications channels.

Computer programs are stored in main memory 205 and/or secondary memory210. Computer programs may also be received via communications interface224. Computer programs, when executed, enable the computer system 200,particularly the processor 204, to implement the methods according tothe present invention. The methods according to the present inventionmay be implemented using software stored in a computer program productand loaded into the computer system 200 using removable storage drive214, hard drive 212 or communications interface 224. The software and/orcomputer system 200 described herein may perform any one of, or anycombination of, the steps of any of the methods presented herein. It isalso contemplated that the methods according to the present inventionmay be performed automatically, or may be invoked by some form of manualintervention.

The invention is also directed to computer products, otherwise referredto as computer program products, to provide software to the computersystem 200. Computer products store software on any computer useablemedium. Such software, when executed, implements the methods accordingto the present invention. Embodiments of the invention employ anycomputer useable medium, known now or in the future. Examples ofcomputer useable mediums include, but are not limited to, primarystorage devices (e.g., any type of random access memory), secondarystorage devices (e.g., hard drives, floppy disks, CD ROMS, ZIP disks,tapes, magnetic storage devices, optical storage devices,Micro-Electro-Mechanical Systems (“MEMS”), nanotechnological storagedevice, etc.), and communication mediums (e.g., wired and wirelesscommunications networks, local area networks, wide area networks,intranets, etc.). It is to be appreciated that the embodiments describedherein can be implemented using software, hardware, firmware, orcombinations thereof.

The computer system 200, or network architecture, of FIG. 2 is providedonly for purposes of illustration, such that the present invention isnot limited to this specific embodiment. It is appreciated that a personskilled in the relevant art knows how to program and implement theinvention using any computer system or network architecture.

The performance of the present invention in terms of imagereconstruction quality was compared with that of several alternativeapproaches, including Channelized Component Analysis (“CCA”), DominantComponent Analysis (“DCA”), Least-Squares Reconstructions (“LESHA” and“LESHAL”) and Multi-Scale Least-Squares Reconstructions (“MULTILES”).For the CCA and DCA methods, the instantaneous amplitude (“IA”) wascomputed using QLM, the instantaneous phase (“IP”) was computed usingthe QEA, the LESHA, LESHAL and MULTILES use QEA to estimate IA and IP.As discussed more fully below, the image reconstruction quality usingRegularized Optimization (“RO”) was superior.

A filterbank covering the whole frequency spectrum consisting of onelow-pass and one high-pass filter is used. Each separable channel filterhas support over four quadrants. To maintain support over only twoquadrants needed for the QEA method, Fast Fourier Transform (“FFT”)pre-filtering is used to remove support in two quadrants, for example,the two left quadrants or two right quadrants. Thus, each bandpassfilter has frequency support in only two quadrants of the frequencyspectrum so that, in effect, each channel filter operates over a singlequadrant. The filters are designed using a min-max, equiripple approach.In a preferred embodiment, passband ripple is set at 0.017 dB and thestopband attenuation is set to 66.02 dB. Because the filterbank coversthe entire spectrum, it can be expected that the instantaneous frequencywill fall within the spectral support of one of the channel filters. Itis assumed that local image coherency will force the instantaneousfrequency estimate to fall within the passband of the dominant bandpassfilter.

FIG. 3, FIG. 4 and FIG. 5 illustrate original input images andreconstructions using the CCA method, the DCA method, and the methodaccording to the present invention. Specifically, FIG. 3 illustrates the“Radial Chirp” image, FIG. 4 illustrates the “Barbara” image and FIG. 4illustrates the “Lena” image. All images contain 512×512 pixels and thesimulation is carried out using Matlab-only code on a 1.83 GHz IntelDual core CPU with a 4G RAM.

Table 1 below is a comparison of the signal to noise ratio of thereconstructed images using DCA, CCA, LESHA, LESHAL, MULTILES and the ROmethod (Equation (2)+DCA) according to the present invention. Valueswith ( )* indicates that the reconstructed image has been normalized.For all cases, the RO approach has superior performance, especially forthe gray scale photograph images Barbara and Lena (see FIG. 4 and FIG.5). The signal-to-noise ratio is greater than 22 dB, compared with themodest signal-to-noise ratio of less than 15.2 than of all other methodsfor all three images tested. In no event is the signal-to-noise ratioless than 15.

TABLE 1 SNR (dB) Image DCA CCA LESHA LESHAL MULTILES (2) + DCA Radial6.51 (14.43)* 3.21 (−2.63)* 0.51 13.56 13.56 15.41 Chirp Barbara 0.92(7.69)* 1.15 (9.76)* 10.48 12.69 12.69 22.39 Lena 0.83 (5.38)* 0.46(6.58)* 14.97 15.16 15.16 24.40

Table 2 is a comparison of the same reconstructed images using astructured similarity index (“SSIM”). SSIM measures the visualstructural similarity between the reconstructed images with the originalreference image. Again, the RO approach (Equation (2)+DCA) offers thehighest SSIM rating, above 0.95 for all three pictures.

TABLE 2 SSIM index [15] Image DCA CCA LESHA LESHAL MULTILES (2) + DCARadial 0.929 0.766 0.144 0.815 0.815 0.978 Chirp Barbara 0.730 0.5440.619 0.656 0.656 0.987 Lena 0.623 0.378 0.731 0.731 0.731 0.962

The visual quality of the reconstructed images is consistent with thequantitative measurements. FIG. 3 is a side-by-side comparison of theoriginal “Radial Chirp” image (FIG. 3A) with images reconstructed usingthe CCA method (FIG. 3B), the DCA method (FIG. 3C), and the RO method(FIG. 3D), respectively. Both the CCA and DCA methods create strongcircular artifacts around the four edges. The contrast of the CCAreconstructed image is noticeably muted. The DCA method introducesdiscontinuous patches into the original continuous image. The RO methodproduced an image without visible artifacts, while maintaining a levelof contrast and continuity comparable to the original image.

The same improved quality holds true for photographic images. FIG. 4 isa side-by-side comparison of the original “Barbara” image (FIG. 4A) withimages reconstructed using the CCA method (FIG. 4B), the DCA method(FIG. 4C), and the RO method (FIG. 4D), respectively. Both the CCA andthe DCA methods create visibly blurry reconstructions with artifacts oflines crossing the image at different angles. Both images lost thetexture detail of the checkered tablecloth and the creases on the pants.The RO reconstruction produces a clean image with the appropriatetexture without visible artifacts. A similar result is achieved in the“Lena” image shown in FIG. 5.

While the disclosure is susceptible to various modifications andalternative forms, specific exemplary embodiments thereof have beenshown by way of example in the drawings and have herein been describedin detail. It should be understood, however, that there is no intent tolimit the disclosure to the particular embodiments disclosed, but on thecontrary, the intention is to cover all modifications, equivalents, andalternatives falling within the scope of the disclosure as defined bythe appended claims.

What is claimed is:
 1. A computer system method for modeling imagecontent comprising the steps of: providing an input image; attaining asmall number of locally coherent components by solving for the minimumof${{J( {a,\zeta} )} = {{\frac{1}{p}{{{f( {a,\zeta} )} - b}}_{p}^{p}} + {\lambda_{a}{T(a)}} + {\lambda_{\zeta}{\zeta }_{1}}}},{{s.t.\mspace{14mu} a} \geq 0},{{\zeta } \leq 1},$wherein J(a,

) is a function that takes vectors as inputs, and whose output is ascalar, a is a one dimensional column vector of image content in termsof an amplitude function,

is a one dimensional column vector of image content in terms of a cosinefunction applied to a phase function, f(a,

) represents a matrix-times-vector notation of the image content interms of the amplitude function and the phase function, b is a onedimensional column vector of the input image approximated by thesummation of a and

, p is a natural and positive number representing the p-norm forfinite-dimensional vector spaces, λ_(a) represents a Lagrange multiplierof the total variation of vector a, T(a) refers to the total variationof vector a, λ

represents a Lagrange multiplier of the vector norm of

, ∥

∥₁ represents the vector norm of

, s. t. represents “such that” the one dimensional column vector a isgreater than or equal to zero and the absolute value of the onedimensional column vector

is less than or equal to one; enforcing a piecewise smooth constrain forone or more instantaneous amplitude functions; calculating a resultingamplitude function and a resulting phase function; reconstructing theinput image using the resulting amplitude function and the resultingphase function to obtain a reconstructed image; and displaying thereconstructed image on a display unit.
 2. The computer system method formodeling image content of claim 1 wherein said attaining step and saidenforcing step are performed simultaneously.
 3. The computer systemmethod for modeling image content of claim 1 wherein said reconstructingstep further comprises the step of utilizing Dominant ComponentAnalysis.
 4. The computer system method for modeling image content ofclaim 1 wherein said reconstructing step further comprises the step ofutilizing Channelized Component Analysis.
 5. The computer system methodfor modeling image content of claim 1 wherein said reconstructing stepuses the cosine of the resulting phase function.
 6. The computer systemmethod for modeling image content of claim 1 wherein the input image isN-dimensional.
 7. The computer system method for modeling image contentof claim 6 wherein the input image is 2-dimensional.